Scientific Part
Mathematics

Injectivity sets of the spherical mean operator with respect to the Bessel convolution

G. V. Krasnoschekikh, Vit. V. Volchkov

Donetsk State University, 24 Universitetskaya St., 283001 Donetsk, Russia

Abstract: Let $C_\natural$ be the set of all even continuous functions on the real axis, $E$ be a non-empty set on $(0,+\infty)$, $\mathcal{R} f(x,t)$ be the spherical mean of the function $f\in C_\natural$ with center at the point $x\in E$ and radius $t>0$ with respect to Bessel convolution. The following problems arise for the operator $\mathcal{R}$: 1) find out whether a given set $E$ is an injectivity set of the transform $\mathcal{R}$; 2) if $E$ is not an injectivity set, then characterize all functions $f\in C_\natural$ such that $\mathcal{R} f(x,t)=0$ on $E\times(0,+\infty)$; 3) if $E$ is an injectivity set, then restore $f$ from the values of $\mathcal{R} f(x,t)$ on $E\times(0,+\infty)$. In this paper we obtain a solution of problems 1 and 2 for an arbitrary set $E\subset(0,+\infty)$, as well as a solution of problem 3 for the case when $E$ is a finite set of injectivity. It is shown that functions from the kernel of the transform $\mathcal{R}$ can be described in terms of series of the eigenfunctions of the Bessel operator converging in the space of distributions. It follows, in particular, that the set $E$ is not the injectivity set of the transform $\mathcal{R}$ if and only if it is contained in the set of zeros of some eigenfunction of the Bessel operator. Moreover, if $E=\{r_1,\ldots,r_m\}$ is a finite injectivity set, we find a class of inversion formula for the transform $\mathcal{R}$ that depend on a set of polynomials $p_1,\ldots,p_m$. It is assumed that $p_1,\ldots,p_m$ have a sufficiently high degree and satisfy some conditions related to the zeroes of Fourier – Bessel transform of Dirac measures with supports at points $r_1, \ldots, r_m$.

Key words: generalised shift, Bessel convolution, Fourier – Bessel transform, spherical means, injectivity sets, inversion formulas.

UDC: 517.444,517.58

Received: 19.01.2025
Accepted: 17.03.2025

DOI: 10.18500/1816-9791-2025-25-4-479-489